In Act III, Scene I of "Twelth Night", Shakespeare writes: "By innocence I swear, And by my youth I have one heart, one bosom and one truth, And that no woman has; nor never none Shall mistress be of it, save I alone." Why does the Bard use a double negative?

Historically, the use of a double negative was used for emphasis.

"A double negative occurs when two forms of negation are used in the same clause. In some languages (or varieties of a language), negative forms are consistently used throughout the sentence to express a single negation. ... Although they are not used in Standard English, double negatives are used in various dialects of English, including Southern American English, African American Vernacular English, and most British regional dialects, most notably the East London (Cockney) and East Anglian dialects. This is similar to negative concord found in other languages, as described later in this article. Often double negatives are considered incorrect grammatical usages; however, dialects which use double negatives do so consistently and follow a different set of descriptive linguistic rules. ... A double negative is also famously used in the first two lines of the song "Another Brick in the Wall (part II)" included in the album The Wall by Pink Floyd, sung by schoolchildren: "We don't need no education, We don't need no thought control". ... Today, the double negative is often considered the mark of an uneducated speaker, but it used to be quite common in English, even in literature."

https://en.wikipedia.org/wiki/Double_negative

For centuries, it was OK to use double and even triple negatives to show how really, really negative something was. Chaucer and Shakespeare did it all the time. (In Twelfth Night, for instance, Shakespeare has Viola using the triple negative “nor never none.”) It wasn’t until the 18th century that the double negative was declared a no-no on the ground that one negative canceled out the other.

If you want your writing to be taken seriously, stay away from examples like “I can’t see nobody” or “He didn’t do nothing.” But a sentence like the one you mentioned (“He’s not unkind”) is perfectly good English. A double negative comes in handy when you want to avoid saying something flatly or hurting somebody’s feelings. Instead of blurting out “Your blind date was a dog,” for example, you might use a double negative to say, “Your blind date was not unattractive.”

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